Welcome to the BA forum on Flyertalk though I am sorry to read of the circumstances.

The ticketing is thanks to Expedia, BA and Qatar have a close relationship and Qatar is one of BA's biggest shareholders. It's true that BKK hasn't been serviced by BA for many years, but that starts again in October. At the moment it would be down to Qatar to fix the flights, since they appear to be the cause of the issue (and overbooking isn't unusual on Qatar). In doing this Expedia should be assisting you in this process, but that may be easier said than done. If you had bought direct from BA then BA would be able to reissue replacement tickets but at the moment it will be Expedia's responsibility to do this, and they will say "get Qatar to do this". Unfortunately if you use an online travel booker using a ticket on codeshare you are at considerable risk of this sort of thing happening. So in short, your staff members should be hassling Qatar over the ticket and you should be giving Expedia hassle to resolve the replacement flights.